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#### Arithmetic Sequences And Their Properties

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### Arithmetic sequences:

An arithmetic sequence is defined as a list of numbers having a definite pattern. Arithmetic sequences are developed such that when the numbers are taken in sequence they can be subtracted by its previous number, the result always remains the same. The difference that exists between all the pairs of consecutive or successive numbers within a sequence is known as the common differences that are usually denoted by the letter “d”. The common difference is used in order to go from one term ti the other. The current term gets added up to the common difference in order to get the next number and this way the sequence carries on. Some of the facts that fall under the arithmetic sequences are as follows:

• When the common difference between two consecutive terms is positive, the sequence is said to be increasing in nature.
• When the common difference between two consecutive terms is negative then the arithmetic sequence is said to be decreasing in nature.

Thus it can be said that arithmetic sequence is a set of numbers which gradually increases or decreases by a constant amount for each term.

The formula for the nth term of a sequence in arithmetic is given in the following form:

an=dn+c

Here d is defined as the common difference

Once the common difference between the numbers is identified, the value of c can be easily plugged in 1 for n where the first term of the sequence is denoted by a1.

Some of the examples of arithmetic sequence re as follows:

1. {1,5,9,13,17,21,25,...}

Since, 5-1= 4

9-5=4 etc

Therefore, the common difference between the numbers is calculated as 4

25 + 4 = 29

29 + 4 = 33

33 + 4 = 37

Hence the next 3 terms will be 29, 33 and 37.

In order to find the nth term we can use the formula for arithmetic sequence while substituting the value by n=1,a1=1n=1,a1=1 and d=4d=4 in the formula of an=dn+c

Using this formula will help in finding c.

Hence, 1=4(1)+c

c=−3

Therefore a formula for the nth term of the arithmetic sequence is

an=4n−3.

2. {12,9,6,3,0,−3,−6,...}

Since 9 -12 = -3

6 – 9 = - 3 and so on hence the common difference of the arithmetic sequence is -3.

It is worthy to be noted here that since the sequence is decreasing in nature the common difference is negative.

In order to find the next 3 terms, it is required to subtract by 3.

-6 – 3 = - 9

-9 – 3 = - 12

-12 – 3 = -15

Hence the next three terms will be -9, -12, and  -15.

Replacing the values we get the desired result of c as below:

12 = -3 (1) + c

c = 15

Hence from the formula, the nth term of the sequence is

an=−3n+15.

### Properties of Arithmetic sequences:

The property of arithmetic sequence is as follow:

1. If a constant gets added to each of the terms of an arithmetic sequence then the resulting sequence is also an arithmetic sequence.

2. If a constant gets subtracted from each of the term of an Arithmetic sequence then the resulting sequence will also be an arithmetic sequence.

3. If each term of an arithmetic sequence is multiplied by a constant then the resulting sequence will also be an arithmetic sequence.

4. If each term of an arithmetic sequence is divided by a constant then the resulting sequence will also be an arithmetic sequence.

5. If the nth term is in linear form then sequence will be in A.P.

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